Chapter 4: Wave equations. What is a wave?. what is a wave?. anything that moves. This idea has guided my research: for matter, just as much as for radiation, in particular light, we must introduce at one and the same time the corpuscle concept and wave concept. Louis de Broglie.
anything that moves
for matter, just as much as for radiation, in particular light, we must introduce at one and the same time the corpuscle concept and wave concept. Louis de Broglie
t1 > t0
t2 > t1
t3 > t2
0 1 2 3
t1 > t0
t2 > t1
t3 > t2
0 1 2 3
To displace f(x) to the right: xx-a, where a >0.
Leta = vt, where v is velocityand tis time
-displacement increases with time, and
-pulse maintains its shape.
So f(x -vt)represents a rightward, or forward, propagating wave.
Now, use the chain rule:
So Þ and Þ
Combine to get the 1D differential wave equation:
works for anything that moves!
at a fixed time:
at a fixed point:
propagation constant: k = 2p/l
wave number:k = 1/l
frequency: n = 1/T
angular frequency:w = 2pn
v = velocity (m/s)
n = frequency (1/s)
v = nl
Thephase, j, is everything inside the sine or cosine (the argument).
y = A sin[k(x ± vt)]
j= k(x ± vt)
For constant phase, dj = 0 = k(dx ± vdt)
which confirms that v is the wave velocity.
A = amplitude
j0 = initial phase angle (or absolute phase) at x = 0 and t =0
The phase velocity is the wavelength/period: v = l/T
Since n = 1/T:
In terms of k, k = 2p/l, and
the angular frequency, w = 2p/T, this is:
v = ln
v = w/k
Here, phase velocity = group velocity (the medium is non-dispersive).
In a dispersive medium, the phase velocity ≠ group velocity.
P = (x,y)
x: real andy: imaginary
P = x + i y
“one of the most remarkable formulas in mathematics”
Links the trigonometric functions and the complex exponential function
eij = cosj + i sinj
so the point,P = Acos(j) + iA sin(j), can also be written:
P = A exp(ij) = Aeiφ
A = Amplitude
j = Phase
Using Euler’s formula, we can describe the wave as
y = Aei(kx-wt)
so that y = Re(y) = A cos(kx – wt)
Math is easier with exponential functions than with trigonometry.
A plane wave’s contours of maximum field, called wavefronts, are planes. They extend over all space.
The wavefronts of a wave sweep along at the speed of light.
Usually we just draw lines; it’s easier.
represents direction of propagation
Consider a snapshot in time, say t = 0:
In complex form:
Y = Aei(k·r - wt)
Substituting the Laplacian operator:
Perpendicular to the direction of propagation:
I. Gauss’ law (in vacuum, no charges):
II. No monopoles:
Perpendicular to each other:
III. Faraday’s law:
Also, at any specified point in time and space,
where c is the velocity of the propagating wave,
Simple case— planewave with E field along x, moving along z:
which has the solution:
Plane of constant , constant E
Energy density (J/m3) in an electrostatic field:
Energy density (J/m3) in an magnetostatic field:
Energy density (J/m3) in an electromagnetic wave is equally divided:
Rate of energy transport: Power (W)
Power per unit area (W/m2):
Take the time average:
A 2D vector field assigns a 2D vector (i.e., anarrow of unit length having a unique direction) to each point in the 2D map.
A light wave has both electric and magnetic 3D vector fields:
A 3D vector field assigns a 3D vector (i.e., anarrow having both direction and length) to each point in 3D space.
You are encouraged to solve all problems in the textbook (Pedrotti3).
The following may be covered in the werkcollege on 14 September 2011:
3, 5, 7, 13, 14,
17, 18, 24